Basic axioms of quantum mechanics

Before describing the axioms of quantum mechanics, one needs some mathematical background in linear vector spaces. Since this may be acquired from any of the introductory textbooks on quantum mechanics, we shall just review some of the main points without going into much detail. 

We use the ket symbol ) introduced by Dirac to denote the state of a physical system. The scalar product (the dot product ) between two vectors is a complex number denoted by 

We can expand the state vector 1 ) in any orthonormal basis of basis vectors, e.g., the eigenstates of the position operator  

Physical observables are represented by Hermitian operators Cl, where a Hermitian operator is defined by Cl = fit. Here Q1 is the so-called adjoint operator defined by the relation 

To identify the expansion coefficients in Eq. (F.3), tp(qi), we multiply from the left with the bra (q-1  

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One postulate that has not explicitly been formulated as a basic axiom of quantum mechanics in the last chapter, because this postulate is valid for any physical theory, is that the equations of quantum mechanics have to be valid and invariant in form in all intertial reference frames. In this chapter, we take the first step toward a relativistic electronic structure theory and start to derive the basic quantum mechanical equation of motion for a single, freely moving electron, which shall obey the principles of relativity outlined in chapter 3. We are looking for a Hamiltonian which keeps Eq. (4.16) invariant in form under Lorentz transformations. 

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